A longstanding open problem is to establish the inviscid limit of classical solutions to the incompressible Navier-Stokes equations for smooth initial data on a domain with boundaries. The question is of great physical and mathematical interest, and it deeply links to the transition to turbulence in fluids that may possibly take place faster than expected due to the presence of a boundary. In this article, I shall give a quick overview of this subject, and then highlight some recent works with my former student, Trinh T. Nguyen, (currently a Van Vleck Assistant Professor at University of Wisconsin, Madison), whose main results establish the inviscid limit for smooth data that are only required to be analytic locally near the boundary. This may be the best possible type of positive results that one can hope for, given the known violent instabilities at the boundary, which I shall discuss below. Before getting on, this picture should already hint at the great delicacy in studying boundary layers (source internet):
to assure the absence of its singular continuous spectrum. More precisely, consider a self-adjoint operator 
(e.g., 
with the usual norm), and assume that there is a self-adjoint operator 
, called a conjugate operator of 
, so that ![\displaystyle P_I i[H,A] P_I \ge \theta_I P_I + P_I K P_I \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+P_I+i%5BH%2CA%5D+P_I+%5Cge+%5Ctheta_I+P_I+%2B+P_I+K+P_I+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0 "\displaystyle P_I i[H,A] P_I \ge \theta_I P_I + P_I K P_I \ \ \ \ \ (1)")
and some compact operator 
on 
denotes the spectral projection of 
, the commutator ![{[H,A] = HA - AH}](https://s0.wp.com/latex.php?latex=%7B%5BH%2CA%5D+%3D+HA+-+AH%7D&bg=ffffff&fg=000000&s=0 "{[H,A] = HA - AH}")
, and the inequality is understood in the sense of self-adjoint operators.
. In his CPAM2000 paper, Grenier proved that there exists no Prandtl’s asymptotic expansion involving one Prandtl’s boundary layer with thickness of order 
, which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order 
. In this paper, we point out how the stability of the classical Prandtl’s layer is linked to the stability of this sublayer. In particular, we prove that the two layers cannot both be nonlinearly stable in 
. That is, either the Prandtl’s layer or the boundary sublayer is nonlinearly unstable in the sup norm.